Equation For Free Surface Of Liquid In Rotating Cylinder At Constant Pressure When R Is Equal To R Calculator

Equation Formula:

\[ Z_s = h_o + \frac{\omega_{Liquid}^2 \times R^2}{4 \times [g]} \]

Height of Free Surface of Liquid without Rotation (h_o):

Angular Velocity of Rotating Liquid (ω_Liquid):

rad/s

Radius of Cylindrical Container (R):

Distance of Free Surface from Bottom of Container (Z_s):

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1. What is the Equation for Free Surface of Liquid in Rotating Cylinder?

The equation calculates the distance of the free surface from the bottom of a cylindrical container when the liquid inside is rotating at constant pressure and the radius is equal to the container's radius. It describes the parabolic shape formed by the liquid surface due to centrifugal forces.

2. How Does the Calculator Work?

The calculator uses the equation:

\[ Z_s = h_o + \frac{\omega_{Liquid}^2 \times R^2}{4 \times [g]} \]

Where:

\( Z_s \) — Distance of free surface from bottom of container (m)
\( h_o \) — Height of free surface of liquid without rotation (m)
\( \omega_{Liquid} \) — Angular velocity of rotating liquid (rad/s)
\( R \) — Radius of cylindrical container (m)
\( [g] \) — Gravitational acceleration constant (9.80665 m/s²)

Explanation: The equation accounts for the parabolic deformation of the liquid surface caused by centrifugal forces when the container rotates about its vertical axis.

3. Importance of Free Surface Calculation

Details: Accurate calculation of the free surface profile is crucial for designing rotating machinery, centrifugal separators, and understanding fluid behavior in rotating systems. It helps predict liquid distribution and potential spillage or dry-out conditions.

4. Using the Calculator

Tips: Enter the height of liquid without rotation in meters, angular velocity in radians per second, and container radius in meters. All values must be non-negative.

5. Frequently Asked Questions (FAQ)

Q1: Why does the liquid surface form a parabola when rotating?
A: The centrifugal force pushes liquid outward, creating a pressure gradient that balances with gravitational force, resulting in a parabolic surface profile.

Q2: What is the significance of constant pressure in this equation?
A: Constant pressure ensures that the free surface is an isobaric surface, meaning all points on the surface experience the same pressure (typically atmospheric).

Q3: Does this equation apply to all rotational speeds?
A: The equation is valid for steady-state rotation where the liquid rotates as a solid body with the container, and surface tension effects are negligible.

Q4: How does container radius affect the free surface profile?
A: Larger radius containers will show more pronounced parabolic deformation at the same angular velocity, as the centrifugal force increases with radius.

Q5: What are the practical applications of this calculation?
A: This calculation is used in designing centrifuges, rotating machinery lubrication systems, spacecraft fuel tanks, and industrial mixing equipment.